The toothless hosts struggled to make the most of their chances against Alan Irvine's men on Saturday, with Jamie Vardy in particular guilty of spurning a couple of good opportunities.
But ultimately they were undone by an error two minutes into the second period when Cambiasso inadvertently directed Wes Morgan's clearance into his own net.
Andy King could have earned Leicester a draw in the final 15 minutes but he headed straight at goalkeeper Ben Foster as Leicester's struggles continued.
West Brom, meanwhile, held on for their first win since the end of September.
Leicester struggled to provide service for their forwards and with top scorer Leonardo Ulloa starting on the bench due to injury, their only real sight of goal in the opening stages saw Danny Drinkwater blaze a shot over from 20 yards.
West Brom – unchanged from their last Premier League outing against Crystal Palace - had to bide their time but after growing into the game they almost grabbed a 17th-minute lead, as the in-form Saido Berahino forced Kasper Schmeichel into a smart save with a shot from the edge of the area.
The hosts were convinced that they should have been awarded a penalty a few moments later, but Stuart Attwell – refereeing his first Premier League game since January 2012 – failed to spot Craig Dawson's handball.
Leicester continued to press and they could have taken the lead in the 37th minute, but Vardy's spectacular overhead kick was straight at Foster.
West Brom almost capitalised on that let-off when Chris Brunt slammed a shot over the crossbar following Berahino's clever cut-back but they were ahead two minutes after the break, courtesy of a helping hand from Cambiasso.
Berahino squeezed a low cross into the area and, although Morgan had seemingly cleared the danger with a stooping header, the ball struck his Argentinian team-mate on the leg and crept inside the post.
Leicester were fortunate not to be dealt a further blow in the 66th minute as Marcin Wasilewski escaped punishment for catching Berahino in the face with his elbow, before King headed straight at Foster with just under quarter-of-an-hour to go.
The hosts did mount a spell of late pressure as they sought an equaliser, but West Brom were largely untroubled.
